Is the QuickDraw with Google neural net a convolutional neural network?
I believe they don't use CNNs. The most important reason why it's because they have more information than a regular image: time. The input they receive is a sequence of (x,y,t) as you draw on the screen, which they refer as "ink". This gives them the construction of the image for free, which a CNN would have to deduce by itself.

They tried two approaches. Their currently most successful approach does the following:


Detect parts of the ink that are candidates of being a character
Use a FeedForward Neural Network to do character recognition on those candidates
Use beam search and a language model to find most the most likely combination of results that results into a word


Their second approach is using an LSTM (a type of Recurrent Neural Network) end-to-end. In their paper they say this was better in a couple languages.

Source: I was an intern in Google's handwriting team in summer 2015 (on which I believe quickdraw is based), but the techniques I explained can be found in this paper.